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Properties

Label 1-4033-4033.179-r0-0-0
Degree $1$
Conductor $4033$
Sign $-0.944 + 0.329i$
Analytic cond. $18.7291$
Root an. cond. $18.7291$
Motivic weight $0$
Arithmetic yes
Rational no
Primitive yes
Self-dual no
Analytic rank $0$

Origins

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Normalization:  

Dirichlet series

L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.286 − 0.957i)3-s + (0.766 + 0.642i)4-s + (−0.998 + 0.0581i)5-s + (−0.597 + 0.802i)6-s + (0.893 + 0.448i)7-s + (−0.5 − 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.957 + 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.835 − 0.549i)12-s + (0.893 + 0.448i)13-s + (−0.686 − 0.727i)14-s + (−0.230 + 0.973i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯
L(s)  = 1  + (−0.939 − 0.342i)2-s + (0.286 − 0.957i)3-s + (0.766 + 0.642i)4-s + (−0.998 + 0.0581i)5-s + (−0.597 + 0.802i)6-s + (0.893 + 0.448i)7-s + (−0.5 − 0.866i)8-s + (−0.835 − 0.549i)9-s + (0.957 + 0.286i)10-s + (−0.957 + 0.286i)11-s + (0.835 − 0.549i)12-s + (0.893 + 0.448i)13-s + (−0.686 − 0.727i)14-s + (−0.230 + 0.973i)15-s + (0.173 + 0.984i)16-s + (−0.173 − 0.984i)17-s + ⋯

Functional equation

\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]
\[\begin{aligned}\Lambda(s)=\mathstrut & 4033 ^{s/2} \, \Gamma_{\R}(s) \, L(s)\cr =\mathstrut & (-0.944 + 0.329i)\, \overline{\Lambda}(1-s) \end{aligned}\]

Invariants

Degree: \(1\)
Conductor: \(4033\)    =    \(37 \cdot 109\)
Sign: $-0.944 + 0.329i$
Analytic conductor: \(18.7291\)
Root analytic conductor: \(18.7291\)
Motivic weight: \(0\)
Rational: no
Arithmetic: yes
Character: $\chi_{4033} (179, \cdot )$
Primitive: yes
Self-dual: no
Analytic rank: \(0\)
Selberg data: \((1,\ 4033,\ (0:\ ),\ -0.944 + 0.329i)\)

Particular Values

\(L(\frac{1}{2})\) \(\approx\) \(-0.08448776906 - 0.4988439557i\)
\(L(\frac12)\) \(\approx\) \(-0.08448776906 - 0.4988439557i\)
\(L(1)\) \(\approx\) \(0.5398298718 - 0.3115112112i\)
\(L(1)\) \(\approx\) \(0.5398298718 - 0.3115112112i\)

Euler product

   \(L(s) = \displaystyle \prod_{p} F_p(p^{-s})^{-1} \)
$p$$F_p(T)$
bad37 \( 1 \)
109 \( 1 \)
good2 \( 1 + (-0.939 - 0.342i)T \)
3 \( 1 + (0.286 - 0.957i)T \)
5 \( 1 + (-0.998 + 0.0581i)T \)
7 \( 1 + (0.893 + 0.448i)T \)
11 \( 1 + (-0.957 + 0.286i)T \)
13 \( 1 + (0.893 + 0.448i)T \)
17 \( 1 + (-0.173 - 0.984i)T \)
19 \( 1 + (0.766 - 0.642i)T \)
23 \( 1 + (-0.766 - 0.642i)T \)
29 \( 1 + (-0.230 - 0.973i)T \)
31 \( 1 + (-0.448 - 0.893i)T \)
41 \( 1 + (-0.866 - 0.5i)T \)
43 \( 1 + (0.342 - 0.939i)T \)
47 \( 1 + (0.116 + 0.993i)T \)
53 \( 1 + (-0.998 + 0.0581i)T \)
59 \( 1 + (0.973 + 0.230i)T \)
61 \( 1 + (-0.116 + 0.993i)T \)
67 \( 1 + (-0.448 - 0.893i)T \)
71 \( 1 + (0.939 - 0.342i)T \)
73 \( 1 + (0.286 + 0.957i)T \)
79 \( 1 + (-0.835 - 0.549i)T \)
83 \( 1 + (0.686 + 0.727i)T \)
89 \( 1 + (0.802 + 0.597i)T \)
97 \( 1 + (0.549 + 0.835i)T \)
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   \(L(s) = \displaystyle\prod_p \ (1 - \alpha_{p}\, p^{-s})^{-1}\)

Imaginary part of the first few zeros on the critical line

−18.71036577573387569681898245878, −18.170885423967583945440141825164, −17.47273920451315220759082146916, −16.62752786083216698018353044184, −16.02555296075731984359058195926, −15.72390489922770290608721439355, −14.93398713813706488438305525837, −14.44858191135891862273379056181, −13.64602928884851721466771976151, −12.56381774148078191279035398384, −11.440282174433307777781050076326, −11.1830287344016584608886417378, −10.4373491950703950056111877062, −10.06204895059852466621103868968, −8.86100529019982366994109402336, −8.383054237917410090418516131065, −7.91460553518610687028654473322, −7.37735394801211661637818840743, −6.16181257410145519354459955189, −5.33298530065959726159805155288, −4.80332673580087893438268687620, −3.60250521977798703543095054951, −3.31838800540859941740992488225, −1.988058606258793494407097996517, −1.11568795066985443865658194654, 0.22218548850022601803435928521, 1.066986786966465533080942596694, 2.12656151132846415338454189356, 2.56018846430570767555028697157, 3.480855304907463436030809897359, 4.39924947464448109076204274434, 5.47045152453010626494605198067, 6.41832673132395984275606958388, 7.28726816748496588529164018955, 7.68445853139077213378335488660, 8.27061765671437396414237765742, 8.85398974863025968163402408701, 9.57496653255108118979070787944, 10.773784973010287482509619193181, 11.279054208340333310731341706387, 11.84064617579820803455287742671, 12.290335640888018931495054609291, 13.22305107382952344604934176218, 13.87424398676033926198089372025, 14.83608280442828041143086053678, 15.59295728482586664142871011028, 15.954281540764452533164333867430, 16.920000874987236939164185517675, 17.79297369316350709275053753624, 18.262537710549904638281150690481

Graph of the $Z$-function along the critical line